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model taking into account sediment deposition
Xiaoping Yuan, Jean Braun, Laure Guerit, Delphine Rouby, Guillaume Cordonnier
To cite this version:
Xiaoping Yuan, Jean Braun, Laure Guerit, Delphine Rouby, Guillaume Cordonnier. A new efficient method to solve the stream power law model taking into account sediment deposition. Journal of Geophysical Research: Earth Surface, American Geophysical Union/Wiley, 2019, 124 (6), pp.1346- 1365. �10.1029/2018JF004867�. �hal-02136641�
A new efficient method to solve the stream power law
1
model taking into account sediment deposition
2
X. P. Yuan1, J. Braun1,2, L. Guerit3, D. Rouby3, and G. Cordonnier4
3
1Helmholtz Centre Potsdam, German Research Centre for Geosciences (GFZ), Potsdam, Germany
4
2Institute of Earth and Environmental Sciences, University of Potsdam, Potsdam, Germany
5
3G´eosciences Environnement Toulouse, UMR5563 CNRS-IRD-Universit´e de Toulouse, France
6
4Ecole polytechnique, Palaiseau, France
7
Key Points:
8
• We present an efficient (O(N) and implicit) method to solve a river erosion model
9
taking into account sediment deposition.
10
• We show how the foreland stratigraphy is controlled by the efficiency of river ero-
11
sion and the efficiency of sediment transport by rivers.
12
• We observe autogenic aggradation and incision cycles in the foreland once the sys-
13
tem reaches a dynamic steady state.
14
Corresponding author: X. P. Yuan,xyuan@gfz-potsdam.de
Abstract
15
The stream power law model has been widely used to represent erosion by rivers, but
16
does not take into account the role played by sediment in modulating erosion and de-
17
position rates. Davy and Lague (2009) provide an approach to address this issue, but
18
it is computationally demanding because the local balance between erosion and depo-
19
sition depends on sediment flux resulting from net upstream erosion. Here, we propose
20
an efficient (i.e.,O(N) and implicit) method to solve their equation. This means that,
21
unlike other methods used to study the complete dynamics of fluvial systems (includ-
22
ing the transition from detachment-limited to transport-limited behavior, for example),
23
our method is unconditionally stable even when large time steps are used. We demon-
24
strate its applicability by performing a range of simulations based on a simple setup com-
25
posed of an uplifting region adjacent to a stable foreland basin. As uplift and erosion
26
progress, the mean elevations of the uplifting relief and the foreland increase, together
27
with the average slope in the foreland. Sediments aggrade in the foreland and prograde
28
to reach the base level where sediments are allowed to leave the system. We show how
29
the topography of the uplifting relief and the stratigraphy of the foreland basin are con-
30
trolled by the efficiency of river erosion and the efficiency of sediment transport by rivers.
31
We observe the formation of a steady-state geometry in the uplifting region, and a dy-
32
namic steady state (i.e., autocyclic aggradation and incision) in the foreland, with aggra-
33
dation and incision thicknesses up to tens of metres.
34
1 Introduction
35
Fixed/Open Boundarybase level
100 km
100 km 5 to 300 km
Closed Boundary erosion
4
3 2
5 6
10
11 9 7
8 1
sediment transport and deposition
foreland uplift
region
(a) (b)
(c)
Figure 1. (a) The concept of source to sink with sediment transport and deposition, modified fromAllen and Heller (2011). (b) Setup for our simulation with uplifting region and foreland basin. (c) Illustration of a simple catchment with normal FastScape stack order (Braun and Wil- lett, 2013). In (b), the red lines indicate the closed boundary where sediment flux cannot leave the system, whereas base level (green line) is fixed as an open boundary.
Quantifying the dynamics of river erosion, sediment transport and deposition (Fig-
36
ure 1a) is a fundamental problem in geomorphology that has great relevance for our un-
37
derstanding of landscape evolution in tectonically active areas. Many parameterizations
38
of these processes have been proposed and implemented in numerical landscape evolu-
39
tion models (Braun and Sambridge, 1997;Chase, 1992;Crave and Davy, 2001; Kooi and
40
Beaumont, 1994;Tucker and Slingerland, 1994).
41
The Stream Power Law (SPL) model has been widely used to represent erosion by rivers (Howard and Kerby, 1983; Whipple and Tucker, 1999). In its simplest form, it as- sumes that erosion rate is proportional to the shear stress exerted by the river on its bed which, in turn, is proportional to net precipitation rate,p, drainage area,A, and local slope,S, according to:
∂h
∂t =U−KfpmAmSn, (1) wherehis topographic elevation,tis time, U is uplift rate,Kf is the fluvial erosion co-
42
efficient, andmandnare the SPL exponents. An important assumption of the SPL model
43
is that sediments are efficiently transported by rivers and not deposited in the simulated
44
domain. The SPL model has been shown to describe a number of fluvial landscapes and
45
processes. It is for example commonly used to infer uplift pattern from river profiles or
46
to model topographic evolution at the scale of a catchment (Braun and Willett, 2013;
47
Campforts et al., 2017;Lav´e and Avouac, 2001). Yet, it is well known that this model
48
might be oversimplified as it does not consider several important processes acting in river
49
channels (Lague, 2014). In particular, it is necessary to take into account the role played
50
by sediment in modulating erosion rate and/or deposition (Whipple and Tucker, 2002),
51
such as a dependence on bedload transport (Davy and Lague, 2009;Kooi and Beaumont,
52
1994) or a bed-cover effect (Cowie et al., 2008;Johnson et al., 2009; Sklar and Dietrich,
53
2001). In fact, transported sediments provide the tools for abrasion and fracturing of rock
54
but also, if overly abundant, they can protect the bedrock from erosion (Sklar and Di-
55
etrich, 1998).
56
Several parameterizations have been proposed to adapt the SPL model to incor-
57
porate the effects of transported sediments, including the erosion-deposition formulation
58
proposed byDavy and Lague (2009), which differs from models based on the divergence
59
of sediment flux (e.g.,Paola and Voller, 2005) in that it conserves mass on the bed and
60
in the water column to treat simultaneous erosion and deposition of a single substrate
61
(Shobe et al., 2017). Based on previous erosion-deposition models (e.g.,Beaumont et al.,
62
1992;Kooi and Beaumont, 1994),Davy and Lague’s (2009) formulation has a limited
63
number of parameters, while attempting to relate these parameters to physical processes
64
and quantities (e.g., saltation length). In addition, their erosion-deposition framework
65
allows the exploration of both detachment-limited and transport-limited models with sim-
66
ple parameter changes, and displays a smooth transition between the two types of model
67
behavior.
68
Davy and Lague’s (2009) formulation has been used or adapted to obtain simple
69
models (e.g.,Carretier et al., 2016;Ganti et al., 2014;Langston and Tucker, 2018;Mouchen´e
70
et al., 2017;Shobe et al., 2017), which assume that the net rate of topographic change
71
is the sum of the erosion rate (controlled by the SPL model) and of the deposition rate,
72
which is proportional to local suspended sediment flux and to a dimensionless deposi-
73
tion coefficient, and inversely proportional to drainage area, a proxy for water discharge.
74
This parameterization is also receiving growing acceptance due to its ability to repro-
75
duce many depositional features of fluvial systems (Carretier et al., 2016;Mouchen´e et al.,
76
2017;Shobe et al., 2017). However, because the local balance between erosion and de-
77
position depends on sediment flux resulting from net upstream erosion, this parameter-
78
ization is computationally demanding.
79
Braun and Willett (2013) have proposed an efficient algorithm for solving the SPL
80
model, which is ideally suited for a large number of model simulations as required for
81
inverting observational constraints in a Bayesian approach. Here we present an equally
82
efficient (i.e.,O(N) and implicit) method to solve the equation proposed byDavy and
83
Lague (2009) that takes into account sediment deposition. O(N) means that the com-
84
putational time increases linearly with the number of points used to discretize the land-
85
scape. An implicit scheme guarantees unconditionally stable time integration of the land-
86
scape evolution equation, which means that large time steps can be used without affect-
87
ing numerical stability. This is potentially an important step as it allows one to use sed-
88
imentological observations, such as the stratigraphy of foreland basins or the position
89
and thickness of river terraces, to further constrain landscape evolution models.
90
Nonetheless, the novelty of our study is not limited to the description of the com-
91
putational efficiency. Based on this algorithm, our model can simulate erosion and de-
92
position in fluvial landscapes, at large spatial (up to thousands of kilometers) and tem-
93
poral (up to tens of millions of years) scales. Because these two processes are considered
94
in one single equation, deposition can occur anywhere in the domain (i.e., not only in
95
depressed areas but also along channels or in stable continental areas). We therefore use
96
our new algorithm to explore the impact of coupling erosion and deposition in a fluvial
97
landscape, and show that this erosion-deposition relationship, which is often ignored, has
98
a strong impact on relief in the uplifted domain, the concavity of the channels and their
99
steepness index. We also demonstrate that this relatively simple model leads to the cyclic
100
formation and destruction of river banks as the stream continuously migrates and some-
101
times erodes into the sediment it has previously deposited. Such autogenic aggradation
102
and incision cycles are currently difficult to simulate in landscape evolution models. The
103
simulations using the new model can thus improve our understanding of the links be-
104
tween external forcings, internal processes, and depositional features.
105
In the next section, we first present our implementation ofDavy and Lague’s (2009)
106
model and theO(N) and implicit numerical scheme. Model implications of our new for-
107
mulation are shown in Section 3. We then explore the model behavior in Section 4 by
108
performing a range of simulations based on a simple setup composed of an uplifting re-
109
gion adjacent to a stable continental area on which a foreland basin develops. In Sec-
110
tion 5, using our model, we observe the formation of autocyclic aggradation and inci-
111
sion in the foreland once the system reaches a dynamic steady state.
112
2 Model implementation
113
2.1 SPL model taking into account sediment deposition
114
The effect of upstream sediment flux was first incorporated into the SPL model by Kooi and Beaumont (1994), by assuming that the rate of topographic change results from the imbalance between a sediment “carrying capacity”,qeqbf , and the upstream sediment yield,qs, according to:
∂h
∂t =U− 1
Lf qfeqb−qs
with qeqbf =K qwS , (2) whereLf is a transport length,K is a dimensionless erosion coefficient,qw = Qw/W is water discharge per unit width (Qw, water discharge, andW, river width), andqsis the sediment flux per unit width obtained by integrating the net upstream erosion rate:
qs= 1 W
Z
A
U −∂h
∂t
dA . (3)
This sediment flux therefore accounts for the whole solid load (bed, suspended, and wash
115
load). The transport lengthLf in (2) can be regarded as the length scale over which the
116
imbalance between the upstream sediment yield and the river carrying capacity is resolved
117
either by deposition (in cases where the river is over-capacity) or by erosion (in cases where
118
the river is under-capacity). Physically,Lf represents the average transport distance of
119
sediment grains within the flow from the location where they are eroded to the location
120
where they are deposited (Beaumont et al., 1992). Lf thus characterizes the proportion
121
of incoming sediment flux which is deposited (the larger the value ofLf, the lower the
122
rate of deposition).
123
Combining equations (2) and (3) leads to:
∂h
∂t =U− 1 Lf
hK QwS W − 1
W Z
A
U−∂h
∂t
dAi
. (4)
To derive the SPL model, two major assumptions are commonly made. Firstly, it is as- sumed that river widthW varies as the square root of the water discharge (Lacey, 1930;
Leopold and Miller, 1956):
W =c Q0.5w , (5)
wherecis an empirical constant, typically of the order of (0.1−1)×10−2 (Montgomery and Gran, 2001, and references therein). Secondly, water discharge can be expressed as the product of net precipitation ratepand drainage areaA:
Qw=pA=p0pA ,˜ (6)
wherep0is mean net precipitation rate, and ˜p = p/p0 represents any spatial or tem- poral variation in precipitationprelative to the mean precipitationp0. Combining equa- tions (5) and (6) with equation (4) leads to
∂h
∂t =U− 1 Lf
hK p0.50
c p˜0.5A0.5S− 1 c p0.50 p˜0.5A0.5
Z
A
U−∂h
∂t dAi
. (7)
The contribution ofDavy and Lague (2009) can be regarded as an improvement onKooi and Beaumont’s (1994) method in an attempt to relate the transport lengthLf
to physical parameters (e.g., water discharge and settling velocity of grains within the flow). According toDavy and Lague’s (2009) formulation, the rate of change of topo- graphic elevation is given by:
∂h
∂t =U −K0qwm0Sn0 + qs Lf
=U−K0qmw0Sn0+d∗vs qw
qs, (8)
whereK0 is an erosion efficiency coefficient,m0 andn0 are two exponents,vsis the net
124
settling velocity of sediment grains, andd∗ is the ratio between the sediment concentra-
125
tion near the riverbed interface and the average concentration over the water column.
126
The value ofd∗(≥1) varies as a function of the Rouse number which defines the rela-
127
tive contribution of bed, suspended, and wash loads (Davy and Lague’s (2009) Figure
128
4). Davy and Lague (2009) discuss howd∗ can be calculated for suspended load and bed
129
load rivers. For small rivers (or large particles), most of the entrainment mechanisms lie
130
in the bed load,d∗ is much larger than 1 and the transport lengthLf is small. Conversely,
131
for large rivers (or small particles) the Rouse number is small,d∗is close to 1 andLf
132
is large. Davy and Lague (2009) present such a model based on the relative contribu-
133
tions of (i) erosion from the bed into the water column (suspended load) and (ii) depo-
134
sition from the water column onto the bed. Thus, the transport lengthLf takes into ac-
135
count the deposition of the bed load and the suspended load. With this approach, the
136
deposition term is proportional to the ratio between the sediment fluxqsand the wa-
137
ter fluxqw. Ifqs qw, deposition is high. On the contrary, ifqs qw, the deposi-
138
tion term tends toward 0. Note that both terms are proportional to the drainage area
139
of the catchment.
140
The equivalent ofKooi and Beaumont’s (1994) transport length Lf in theDavy and Lague’s (2009) approach is therefore:
Lf = qw d∗vs
= Qw W d∗vs
= pA
W d∗vs
= p0pA˜ W d∗vs
= pA˜
W G, with G= d∗vs p0
, (9)
whereGis a dimensionless deposition coefficient, which is a function of the sediment con-
141
centration ratio in transport, the settling velocity of sediment, and the mean precipita-
142
tion rate. G/p˜is identical to Θ as defined inDavy and Lague (2009). They showed that,
143
for typical values of the various parameters (p= 10−7 m/s,vs∈
10−6−10−1 m/s),
144
G/˜pis of order 1 or greater, in good agreement with estimates from natural sedimen-
145
tary systems (Guerit et al.(2018), pers. comm.).
146
Equations (3), (5), (6) and (9) can be combined with equation (8) to yield:
∂h
∂t =U−K0Qw W
m0
Sn0+ 1 W Lf
Z
A
U−∂h
∂t
dA
=U−K0p0.5m0 0
cm0 p˜0.5m0A0.5m0Sn0+ G
˜ pA
Z
A
U−∂h
∂t
dA .
(10)
In the parametric study ofDavy and Lague (2009),m0 =n0 = 1 and their erosion term
147
K0p0.50
c p˜0.5A0.5S is thus similar to that ofKooi and Beaumont (1994). The main differ-
148
ences between their models are: (i) the depositional term is inversely proportional to ei-
149
ther the drainage area (Davy and Lague, 2009) or the square root of drainage area (Kooi
150
and Beaumont, 1994); and (ii)Lf is applied only toqs(Davy and Lague, 2009), or is ap-
151
plied to bothqeqbf andqs (Kooi and Beaumont, 1994).
152
ReplacingK0p0.50 /cm0, 0.5m0 andn0 in (10) byKf,mandn, respectively, we can make a more direct connection to the SPL model and write that the rate of topographic change∂h/∂tin response to tectonic uplift, river erosion and sediment deposition is given by:
∂h
∂t =U−Kfp˜mAmSn+ G
˜ pA
Z
A
U−∂h
∂t
dA . (11)
The modified SPL formulation has only one additional parameter (i.e., the dimension-
153
less deposition coefficientG) compared to the classic SPL model. The dimensionless con-
154
stantGmultiplying the deposition rate in (11) depends on an assumed mean precipi-
155
tation rate; any spatial or temporal variation in precipitation rate is introduced through
156
the variable ˜p. At steady state, the catchment areaAin the deposition term vanishes,
157
and the deposition term is equal toGU/˜p. Note that this modified SPL formulation is
158
constructed by considering fluxes and, therefore, it does not specifically consider the ef-
159
fect of grain size. However, the mathematical definition ofGin (9) makes it related to
160
the size of the sediments in transport through the settling velocityvs. In this work, we
161
consider different values ofG=d∗vs/p0 as a whole, rather than studying individually
162
different values ofd∗,vs, andp0.
163
Fluvial erosion leads to the formation of hillslopes along river channels. Fluvial ero- sion and hillslope processes are interdependent, therefore hillslope processes need to be included in our model, which are commonly represented by a linear diffusion term (Ah- nert, 1967):
∂h
∂t =Kd∇2h , (12)
whereKd is a hillslope sediment transport coefficient. In our model, the diffusion equa-
164
tion (12) is calculated separately, after solving equation (11). Both equations are applied
165
in every cell of the landscape.
166
Easily detachable materials such as unconsolidated sediments should be character-
167
ized by a larger erosion coefficientKf than bedrock (Davy and Lague, 2009;Kooi and
168
Beaumont, 1994). Therefore,Kf depends on whether topographic elevationhis higher
169
than basement elevationhbase or not. In areas that are in net erosion (i.e.,h≤hbase),
170
we assumeKf =Kf b (subscriptb represents bedrock), whereas in areas covered by sed-
171
iments (i.e.,h > hbase), we assumeK = Kf s(subscripts represents sediments). In
172
most of our simulations, we assumeKf b = Kf sfor the sake of simplicity, and we use
173
Kf b 6=Kf sfor our sensitivity analysis in Section 4.2.2.
174
2.2 O(N) and implicit algorithm
175
The most challenging part is to solve equation (11) in an efficient manner (i.e., in O(N) operations) and using an implicit algorithm that allows for large time steps. For this we first discretize equation (11) using a backward Euler implicit finite difference scheme for each of thenx×ny nodes (nx andny are the number of nodes to discretize the land- scape in thex- andy-directions, respectively) as follows:
ht+∆ti −hti
∆t =U−Kfp˜mAmi ht+∆ti −ht+∆trec(i)
∆li
n
+ G
˜ pA˜i
X
j=ups(i)
U−ht+∆tj −htj
∆t
, 1≤i≤nx×ny, (13)
in which,hti andht+∆ti are the elevations of thei-th node at timetand time t+∆t, re- spectively,ht+∆trec(i)is the elevation of the i-th node’s receiver (the node in the steepest- descent drainage direction of thei-th node) at time t+∆t, ∆li is the distance between thei-th node and its receiver, andP
j=ups(i) represents the sum of thei-th node’s up- stream catchment nodes. ˜Ai in equation (13) is a dimensionless catchment area defined as:
A˜i=Ai/(∆x∆y) =Ni, (14) where ∆x,∆y are the horizontal sizes of the cells, andNi is simply the number of cells
176
upstream of celli. To compute the catchment areasAi in O(N) operations, we use the
177
reverse stack order as defined in the FastScape algorithm (Braun and Willett, 2013).
178
To explain the remaining parts of our proposed numerical scheme, we assume that n = 1. The general case (n 6= 1) is dealt with later. Whenn = 1, equation (13) can be expressed as
−Fiht+∆trec(i)+ (1 +Fi)ht+∆ti + G
˜ pA˜i
X
j=ups(i)
ht+∆tj =bti, with Fi= Kfp˜mAmi ∆t
∆li , and bti =hti+U∆t+ G
˜ pA˜i
X
j=ups(i)
(htj+U∆t).
(15)
The termbti on the right-hand side of the equation is known from the solution at time
179
t, while elevations on the left-hand side are unknown at timet+ ∆t.
180
For the nodes at base level (open boundary, Figure 1b), we assume that the ele- vation is constant through time:
ht+∆tbase level =htbase level. (16) We also assume that sediment can leave the system from these base level nodes.
181
The above finite difference equations can be expressed in the following matrix form:
B·ht+∆t=bt. (17)
As shown in Appendix A, if we use the FastScape stack order in Figure 1c to solve equa- tion (17), every rowi ofB has a single non-zero element before the diagonal element that corresponds to the receiver of nodei, and many non-zero elements after the diagonal el- ement that correspond to all upstream nodes ofi. Solving (17) by factorizing the ma- trixB (e.g., by Gauss-Jordan elimination) is a problem of complexity ofO(n3). To ob- tain a greater efficiency, we use a Gauss-Seidel iteration scheme to computeht+∆t in equa- tion (17). This iterative algorithm requires to split the matrixB into its lowerF and strictly upper triangular matrixE as follows:
B·ht+∆t= (F+E)·ht+∆t=bt, (18)
3 4 5 6 log
10(n
x× n
y)
0 2 4 6 8 10 12
Number of iterations
G = 0
G = 0.01 G = 0.1
G = 1 G = 2
Figure 2. The number of iterations required for the solution to be convergent as a func- tion of resolutionnx × ny. Solving equation (19) on a square area of resolutionnx × ny = 961,10000,99856, and 1000000, usingU = 0.2 mm/yr,Kf = 2×10−5 m1−2m/yr,p= 1 m/yr, and varying the value ofG.
where the matricesF andE are shown in Appendix A.
182
The Gauss-Seidel iterative process starts with an initial guessht+∆t,0 =htand uses the following recurrence to obtain an improved estimateht+∆t, k+1:
F·ht+∆t, k+1=bt−E·ht+∆t, k, (19)
from the valueht+∆t, k obtained at the previous iteration. Interestingly, equation (19) can be written in a different form for each nodei:
ht+∆t, k+1i =bti−G/(˜pA˜i)P
j=ups(i)ht+∆t,kj +Fiht+∆t, k+1rec(i)
1 +Fi , (20)
if the nodes are processed in the FastScape stack order.
183
The procedure is continued until the maximum difference in node elevation between two successive iterations is below a given tolerance(expressed in meters) as
max|ht+∆t,k+1i −ht+∆t,ki |< for allisuch that 1≤i≤nx×ny. (21) The tolerance is taken as a small fraction (10−3) of the increment in topography,U∆t.
184
The above procedure based on a Gauss-Seidel iterative scheme is known to con-
185
verge if the matrixB is strictly diagonally dominant: | −Fi|+|P
j=ups(i)G/(˜pA˜i)|<
186
|1+Fi|, thusG/˜p < Ni/(Ni−1) after some derivations. Therefore, the iterative method
187
is proven to converge unconditionally at least whenG/˜p≤1, but we show experimen-
188
tally in section 4 that this method can also converge even if this condition is not satis-
189
fied.
190
As shown in Figure 2, our new implicit method to solve equation (13) isO(N) as
191
the number of iterations required in the Gauss-Siedel scheme depends on the value of
192
Gbut not on the resolution of the model (nx×ny).
193
We note that the left-hand side of equation (19) is the same as in the FastScape
194
algorithm (Braun and Willett, 2013) while the right-hand side only differs by a single
195
term given by: E·ht+∆t, k at time t+ ∆t. The implementation of our new algorithm
196
is therefore a very simple addition to the FastScape algorithm. Note also that, if the value
197
of the deposition coefficientGis null, the right-hand side term simplifies tobt, and the
198
new algorithm is identical to the basic FastScape algorithm which does not require the
199
Gauss-Seidel iteration to obtain the elevation at timet+ ∆t.
200
The above Gauss-Seidel iteration algorithm can be extended to consider different values ofKf andGto differentiate between bedrock and previously deposited sediments during the iteration process. Basement elevationht+∆tbase at each step is obtained using:
ht+∆tbase = min(ht+∆t, htbase+U∆t), (22) wherehtbase+U∆t is the basement elevation resulting from uplift without surface pro-
201
cesses.
202
When the value ofnis not equal to 1, equation (13) becomes:
ht+∆t, k+1i +Kfp˜mAmi ∆tht+∆t, k+1i −ht+∆t, k+1rec(i)
∆li
n
=hti+U∆t+ G
˜ pA˜i
X
j=ups(i)
(htj+U∆t−ht+∆t, kj ). (23)
This non-linear equation can be solved by combining the Gauss-Seidel iterations with
203
a local Newton scheme. We solve the diffusion equation (12) using an alternating direc-
204
tion implicit andO(N) scheme (Peaceman and Rachford, 1955).
205
3 Model implications on geomorphological relationships
206
Before studying the behaviour of the numerical scheme presented above, we first
207
wish to derive several basic geomorphological relationships from the evolution equation
208
(11). These include the steady-state slope-area relationship, the shape of steady-state
209
river profile, and the expression for the response time (i.e., the time necessary to reach
210
steady state). In this section, all relationships are for the uplifting region only, and we
211
neglect the hillslope processes.
212
3.1 Steady-state slope-area (S−A) relationship
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At steady state (i.e., when uplift is balanced by channel incision assumed to be gov- erned by the SPL model), the slope and intercept of the relationship between slope and drainage area provide constraints on the concavity (the ratiom/n) and the steepness in- dex (ks= (U/Kf)1/np˜−m/n), respectively (e.g.,Wobus et al., 2006, and references therein).
When taking deposition into account, the SPL model must be replaced by equation (11), which, under the assumption of steady state (i.e., ∂h∂t = 0) leads to the following slope- area (S−A) relationship:
S=h(1 +G/p)U˜ Kf
i1/n
A−m/np˜−m/n= (1 +G/p)˜1/n(U/Kf)1/nA−m/np˜−m/n. (24) From this equation, a new steepness index can be defined:
k0s= (1 +G/˜p)1/n(U/Kf)1/np˜−m/n, (25) that only differs by a factor (1 +G/˜p)1/n. For typical values ofG/˜p = 1 (Davy and
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Lague, 2009) andn= 1 (Whipple and Tucker, 1999), the effect of sediment deposition
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is to increase the steepness index by a factor of 2.
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3.2 Steady-state river profile
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At steady state, a power-law relationship is generally observed between the length of a stream,x, and its upstream drainage area (Hack, 1957;Lague et al., 2003;Mont- gomery and Dietrich, 1992;Morisawa, 1962;Walcott and Summerfield, 2009) that can
be expressed as:
A=k xb, (26)
wherekandb, usually close to 0.5 and 2, respectively, are called Hack’s law coefficients.
Combining equation (26) with equation (24) leads to:
S=dh
dx = (1 +G/˜p)1/nU1/nKf−1/np˜−m/nk−m/nx−bmn . (27) By integrating the above equation over the length of the stream, we obtain the steady- state river profile:
h(x) = (1 +G/p)˜1/nU1/nKf−1/np˜−m/nk−m/n(1−bm
n )−1x1−bmn +C , for bm6=n , (28) whereC= 0 if the elevation at base level is zero (i.e.,h(0) = 0). We see that the steady-
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state elevation of every point along the river profile is (1 +G/˜p)1/ntimes higher than
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what is expected from the classic SPL model (i.e., without sediment deposition).
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3.3 Response time
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The response time of a fluvial landscape can be defined as the time needed for river profiles to reach steady state. In our model, the response time is
τ0=h(L)/U = (1 +G/p)˜1/nU1/n−1Kf−1/np˜−m/nk−m/n(1−bm
n )−1L1−bmn , (29) whereLis the length of the uplifting region. In the classic SPL form (without the term of deposition), the response time of a fluvial landscape is (Whipple, 2001)
τ=U1/n−1Kf−1/np˜−m/nk−m/n(1−bm
n )−1L1−bmn . (30) The ratio between these two response times is once again given by:
τ0
τ = (1 +G/˜p)1/n. (31)
The impact of continental deposition on the evolution of a fluvial landscape is highlighted by the three geomorphological relationships above: steady-state slope is higher, average topography in the uplifting region is higher, and response time is longer than with the classic SPL model (i.e., without deposition). However, neither the shape of the river pro- file, nor the dependency of the response time or the steepness index to other parame- ters (such asKf,m andn, or the length of the channel, precipitation rate or the uplift rate) are affected by sediment deposition. The dimensionless deposition coefficientGalone controls the difference with respect to the classic SPL model and appears only inside a multiplying factor, (1 +G/p)˜1/n. In all three relationships we derived above, the fac- tor (1 +G/p)˜1/nmultiplies the other poorly constrain factorKf−1/n; this means that the effect of sediment deposition (on most morphometric measures and scales) can be included in the SPL erosion coefficientKf by simply redefining this constant in the fol- lowing way:
Kf0 =Kf/(1 +G/p)˜ . (32) This also means that the value of the constantGcannot be easily derived from the con-
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cavity of rivers, the total relief of river channels or the response time of fluvial erosion.
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4 Model behavior
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We now demonstrate the behavior and applicability of our numerical implemen-
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tation of the modified SPL model by performing a range of simulations. These simula-
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tions are based on a simple setup composed of an uplifting region adjacent to a stable
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Table 1. Parameters for the simulations.
Notation Definition Values/Range Unit
h elevation of topography m
hbase elevation of basement m
t time 10 Myr
∆t incremental time 1000 yr
x horizontal dimension 105 to 400 km
y horizontal dimension 100 km
∆x,∆y cell size 1 km
U uplift rate 0.2, 0.5, 1 mm/yr
p net precipitation rate 0.5, 1, 2 m/yr
p0 mean net precipitation rate 1 m/yr
˜
p=p/p0 variation ratio of precipitation rate 0.5, 1, 2 -
A drainage area m2
S surface slope in drainage direction -
m SPL exponent 0.4a -
n SPL exponent 1b -
Kf b SPL erosion coefficient of bedrock 2×10−5c m1−2m/yr Kf s SPL erosion coefficient of sediments (2,4,8)×10−5 m1−2m/yr
G deposition coefficient (0,0.1,1,10)d -
Kd hillslope diffusion coefficient 0.01e m2/yr
aParameters fromStock and Montgomery(1999) andPerron et al.(2009);bparameters fromBraun and Willett (2013);Stock and Montgomery(1999);Whipple and Tucker(1999) andBraun and Willett(2013);
cparameters fromWhipple and Tucker (1999);dparameters fromDavy and Lague(2009);eparameters fromDensmore et al.(2007) andArmitage et al.(2013).
continental area on which a foreland basin is allowed to develop. Setup for the simula-
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tions is a rectangular area of (105 to 400)×100 km2 (Table 1, Figure 1b). The initial to-
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pography has random noise elevation of up to 1 m. This domain is discretized into a num-
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ber of cells with a cell size of ∆x= ∆y= 1 km. The left-hand side of the domain (100×
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100 km2) is uplifted at a constant rateU while the foreland is fixed, and the foreland
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edge (base level) through which the sediments can leave the system is also fixed. We use
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a constant diffusion coefficient ofKd = 0.01 m2/yr in all simulations. The net mean
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precipitation ratep0 is set to 1 m/yr. We perform a series of model runs for a total time
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of 10 Myr (10 000 time steps of duration 1000 yr) (Table 1).
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4.1 Simulations without foreland
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We first performed a series of simulations without the foreland to better illustrate
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the effect of sediment deposition. We run a simulation withU = 0.2 mm/yr,Kf = 2×
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10−5 m0.2/yr, p= 1 m/yr andG= 0 to test the classic SPL model without sediment
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deposition. During the experiment, topography rises, and after 4 Myr, reaches steady
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state. The final simulated landscape is shown in Figure 3a and the evolution of the mean
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elevation is presented in Figure 3b. As observed from the black curve (G= 0, Figure
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3b), the mean elevation in the uplifting region increases progressively before decreasing,
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and then reaches a constant value at steady state. The same behavior is observed inDavy
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and Lague’s (2009) detachment-limited simulations (their Figure 1). Note that this non-
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monotonous evolution of the mean elevation is probably related to the presence of lo-
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cal minima which are important at the beginning because we start with an initial topog-
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raphy that has random noise. Local minima are known to artificially reduce the erosional
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